In [Inv. Math., 1978], Morgan proved that almost every curve in R^3 is the boundary of a unique area minimizing surface. I will show how to extend Morgan's result to surfaces of any dimension and codimension. The result follows from the generic existence of boundary points with density 1/2, which exploits a boundary regularity theorem recently proved by De Lellis, De Philippis, Hirsch and Massaccesi. The argument to deduce the generic uniqueness combines a general unique continuation principle with Almgren's celebrated regularity theory, ensuring that the singular set of any area minimizing current has codimension at least two and therefore it cannot disconnect the regular part. I will then explain how to prove a generic uniqueness result for a Plateau-type problem studied in optimal transport, even if the singular set has only codimension one.